Locally strongly convex affine hyperspheres realizing Chen's equality
Locally strongly convex affine hyperspheres realizing Chen's equality
复制标题
实现 Chen 等式的局部强凸仿射超球面
DOI：
10.1016/j.jmaa.2017.07.073
复制
发表时间：
2017
影响因子：
1.3
通讯作者：
Xu Huiyang
中科院分区：
文献类型：
--
作者：
Li Cece;Xu Huiyang
在超曲面的仿射微分几何中，C。Scharlach等人发现了一个包含内在曲率和外在曲率的不等式，并将椭圆和双曲仿射超球面分类，如果仿射不变的二维分布D2是可积的，则实现等式。本文继续研究实现等式的仿射超球，包括抛物仿射超球。作为主要结果，首先我们对抛物仿射超球面进行了分类，如果其数量曲率为常数，或者D2是可积的，则它们实现相等。其次，当D2不可积时，通过引入一个定义良好的三维分布D3，我们完成了局部强凸仿射超球面的分类，实现了D3可积时的等价性。最后，我们提出了一个猜想和一个问题，以确定所有仿射超球面达到平等。
In affine differential geometry of hypersurface, C. Scharlach et al. found an inequality involving intrinsic and extrinsic curvatures, and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2-dimensional distribution D2 is integrable. In this paper, we continue to study affine hyperspheres realizing the equality, including parabolic affine hyperspheres. As main results, firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant, or D2 is integrable. Next, by introducing a welldefined 3-dimensional distribution D3 when D2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality.